12 T. H. Havelock. 
By taking points between the bow and stern and behind the stern, it may be 
verified that (41) gives this part of the elevation for all points on the under- 
standing that each function Q, is symmetrical about the zero of its argument 
while each function Q, is anti-symmetrical, that is Q)(— vu) = Q, (wu) and 
Q, (— 4) = — Q, (w). 
In (40) and (42) we have the total elevation expressed in terms localised 
at the bow and stern, and in functions which are easily calculated and tabulated. 
The quantities have been calculated, without attempting any great degree of 
accuracy, but sufficiently to show the character of the curves. These are shown 
in fig. 3 in relation to the length of the model for the velocity given by 
Kol = gl/u? = a. 
Fie. 3. « 
The total elevation is the sum of the four curves which are shown in fig. 3. 
One curve, symmetrical fore and aft, is the complete local disturbance given 
by the sum of all the Q terms. Then there are two equal curves, one starting 
at the bowand the other at the stern, for the wave terms due to the discontinuity 
in slope at the bow and stern. The fourth curve is the total contribution of 
the curved surface to the wave part of the elevation. 
9. Another case of interest, which will only be mentioned here, is an un- 
symmetrical model whose wave resistance has been discussed previously ; 
its form is given by 
y=—ah(h+l?, —l<h<0. (43) 
Here there is only one discontinuity in f’ (h), namely, at the bow, and f” (h) is a 
linear function of h throughout the range. It will be found that the wave 
elevation requires the first three terms Pp», P,, P, in the series of P functions, 
while the local disturbance can be expressed in terms of Qo, Q,, Q of a similar 
series of Q functions. 
To return to the general expression (35), it will be seen from the examples 
that the localisation of the disturbance at special points is largely a matter of 
